Comment by abetusk

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Comment by abetusk

Power law distributions are specializations of the more general Levy stable distributions [0] [1]. Levy stable distributions answer the following question:

Given that the sum of independent and identically distributed random variables that converge to a distribution, what is the distribution they converge to?

If you answered Gaussian, you'd be wrong. The correct answer is Levy stable. There was no condition on finite variance. When variance can be infinite, Levy stable, or power law tail distributions, is the result. When the variance is finite, a Gaussian is the limiting distribution and, consequently, a Gaussian distribution is part of the family of Levy stable distributions.

The stability quality is the reason why the Levy stable (aka power law tail) distributions show up all over the place. If you've ever heard that the reason why the Normal distribution is called "normal", because the sums of (finite variance) random variables converges to a Gaussian, the same reasoning applies to the Levy stable. In some sense, Levy stable distributions are more normal than the normal distribution. My opinion is that infinite variance is hard for people to wrap their heads around, so they reject the premise.

Unfortunately I don't have a good answer for what the article brings up about the Fourier transform, but I'm almost positive that this can be answered with Levy stable distributions in mind. I will say that the distribution is often characterized by it's characteristic function. A short perusal of Wikipedia talks about Levy stable distributions being closed under Fourier transforms, which is what the article is talking about.

[0] https://en.wikipedia.org/wiki/L%C3%A9vy_distribution

[1] https://en.wikipedia.org/wiki/Stable_distribution

Replies abetusk · 2026-08-10
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